The FO model-checking conjecture for monadically dependent hereditary graph classes

Let $\mathcal{C}$ be a hereditary class of graphs. FO model-checking conjecture. There is an FPT first-order model-checking algorithm for graphs in $\mathcal{C}$ if and only if $\mathcal{C}$ is monadically dependent. This is described as a major conjecture in finite model theory; the equivalence is presented as the unresolved implication underlying the paper's dichotomy, while the surrounding discussion also invokes the assumption ${\mathsf{FPT}}\neq {\mathsf{AW}[*]}$ for the corresponding algorithmic characterization.